Understanding deep neural networks (DNNs) is a key challenge in the theory of
machine learning, with potential applications to the many fields where DNNs
have been successfully used. This article presents a scaling limit for a DNN
being trained by stochastic gradient descent. Our networks have a fixed (but
arbitrary) number $L\geq 2$ of inner layers; $N\gg 1$ neurons per layer; full
connections between layers; and fixed weights (or "random features" that are
not trained) near the input and output. Our results describe the evolution of
the DNN during training in the limit when $N\to +\infty$, which we relate to a
mean field model of McKean-Vlasov type. Specifically, we show that network
weights are approximated by certain "ideal particles" whose distribution and
dependencies are described by the mean-field model. A key part of the proof is
to show existence and uniqueness for our McKean-Vlasov problem, which does not
seem to be amenable to existing theory. Our paper extends previous work on the
$L=1$ case by Mei, Montanari and Nguyen; Rotskoff and Vanden-Eijnden; and
Sirignano and Spiliopoulos. We also complement recent independent work on $L>1$
by Sirignano and Spiliopoulos (who consider a less natural scaling limit) and
Nguyen (who nonrigorously derives similar results).

本文提出了一种适用于深度神经网络的缩放极限的解决方案，其权重可由被描述为平均场模型的理想粒子近似表示，该问题的关键在于我们的 McKean-Vlasov 问题存在唯一解。

某些深度神经网络的平均场极限

A mean-field limit for certain deep neural networks

The purpose of these notes, based on a course given by the second author at
Les Houches summer school, is to explain the probabilistic construction of
Polyakov's Liouville quantum gravity using the theory of Gaussian
multiplicative chaos. In particular, these notes contain a detailed description
of the so-called Liouville measures of the theory and their conjectured
relation to the scaling limit of large planar maps properly embedded in the
sphere. These notes are rather short and require no prior knowledge on the
topic.

这篇文章主要介绍了通过高斯乘性混沌理论来进行概率建模，从而实现 Polyakov's Liouville 量子引力模型的构建，详细描述了所谓 Liouville 测度的理论以及它们与球形嵌入大型平面图的比例极限的可能关系。